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How should I find the derivative of y = x^(e^x)?

I tried using the chain rule along with the power rule, coming out to:

(e^x) (e^x) (X^(e^x - 1))

If I had took the natural log of both sides and then used implicit differentiation, I would have gotten as a derivative:

(x^(e^x)) (e^x) (1/x + ln x)

which is the correct answer according to my TI89.

Why wouldn't the first method work? Or was there any flaw?

By the way I just started Calculus as a high schooler.

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# Finding derivative question

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